Hi, I'm having trouble to find how write moments of random variables in symbolic terms. For instance, say I define z = x + y then I want to compute the variance of z in terms of moments of x and y. I would like Mathematica to return this:
Variance(z) = variance(x) + variance(y) + 2*covariance(x,y)
(All in symbolic terms)
My actual problem is a bit more complicated. I have that the data generating process of a scalar random variable X is an AR(p):
$X_{t} = \sum_{j=1}^{p} A_{j} X_{t-j} + u_{t}$
where $A_{j}$'s are scalars, for some white noise $u_{t}$. I don't want to specify a distribution for $u_{t}$, and there is NO data nor estimation involved. This is a population exercise.
I want to compute the variance of $X_{t}$ as a function of $p$, the $A_{j}$'s, and probably the variance of $u_{t}$. The process is stationary.
MY QUESTION: can mathematica give me the expression of Variance( $X_{t}$ ) in symbolic terms?
Thanks!!!! All help/comments welcome!
PS Just in case I'm familiar with the well known formula for it, but I don't want to write that directly. I want mathematic to tell me the expression. The reason is that I also have more complicated processes to look into where deriving the variance is a mess, so I would like mathematica to do it for me. For instance, I have VAR(p) processes for X and Y.